By defining a new terminology, scatter degree, as the supremum of graininess functional value, this paper studies the existence of solutions for a nonlinear two-point dynamic boundary value problem on time scales. We do not need any growth restrictions on nonlinear term of dynamic equation besides a barrier strips condition. The main tool in this paper is the induction principle on time scales.

Calculus on time scales, which unify continuous and discrete analysis, is now still an active area of research. We refer the reader to [1–5] and the references therein for introduction on this theory. In recent years, there has been much attention focused on the existence and multiplicity of solutions or positive solutions for dynamic boundary value problems on time scales. See [6–17] for some of them. Under various growth restrictions on nonlinear term of dynamic equation, many authors have obtained many excellent results for the above problem by using Topological degree theory, fixed-point theorems on cone, bifurcation theory, and so on.

In 2004, Ma and Luo [18] firstly obtained the existence of solutions for the dynamic boundary value problems on time scales

(1.1)

under a barrier strips condition. A barrier strip is defined as follows. There are pairs (two or four) of suitable constants such that nonlinear term does not change its sign on sets of the form , where is a nonnegative constant, and is a closed interval bounded by some pairs of constants, mentioned above.

The idea in [18] was from Kelevedjiev [19], in which discussions were for boundary value problems of ordinary differential equation. This paper studies the existence of solutions for the nonlinear two-point dynamic boundary value problem on time scales

(1.2)

where is a bounded time scale with ,, and . We obtain the existence of at least one solution to problem (1.2) without any growth restrictions on but an existence assumption of barrier strips. Our proof is based upon the well-known Leray-Schauder principle and the induction principle on time scales.

The time scale-related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to time scales. Here, in order to make this paper read easily, we recall some necessary definitions here.

A time scale is a nonempty closed subset of ; assume that has the topology that it inherits from the standard topology on . Define the forward and backward jump operators by

(1.3)

In this definition we put ,. Set ,. The sets and which are derived from the time scale are as follows:

(1.4)

Denote interval on by .

Definition 1.1.

If is a function and , then the delta derivative of at the point is defined to be the number (provided it exists) with the property that, for each , there is a neighborhood of such that

(1.5)

for all . The function is called -differentiable on if exists for all .

Parallel to the definition of delta derivative, the notion of nabla derivative was introduced, and the main relations between the two operations were studied in [7]. Applying to the dual version of the induction principle on time scales (Remark 1.6), we can obtain the following result.

Theorem 3.1.

Let be continuous. Suppose that there are constants , , with , satisfying

(S1)

, ,

(S2)

for , for ,

where

(3.1)

Then dynamic boundary value problem

(3.2)

has at least one solution.

Remark 3.2.

According to Theorem 3.1, the dynamic boundary value problem related to the nabla derivative

Acknowledgments

H. Luo was supported by China Postdoctoral Fund (no. 20100481239), the NSFC Young Item (no. 70901016), HSSF of Ministry of Education of China (no. 09YJA790028), Program for Innovative Research Team of Liaoning Educational Committee (no. 2008T054), and Innovation Method Fund of China (no. 2009IM010400-1-39). Y. An was supported by 11YZ225 and YJ2009-16 (A06/1020K096019).

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